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Permanent indentation characterization for low-velocity
impact modelling using three-point bending test
Natthawat Hongkarnjanakul, Samuel Rivallant, Christophe Bouvet, Arturo
Miranda
To cite this version:
Natthawat Hongkarnjanakul, Samuel Rivallant, Christophe Bouvet, Arturo Miranda. Permanent
in-dentation characterization for low-velocity impact modelling using three-point bending test. Journal of
Composite Materials, SAGE Publications, 2014, 48 (20), pp.2441-2454. �10.1177/0021998313499197�.
�hal-01851599�
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Eprints ID: 9323
To link to this article: DOI: 10.1177/0021998313499197
URL:
http://dx.doi.org/10.1177/0021998313499197
To cite this version:
Hongkarnjanakul, Natthawat and Rivallant, Samuel
and Bouvet, Christophe and Miranda, Arturo Permanent indentation
characterization for low-velocity impact modelling using three-point
bending test. (2013) Journal of Composite Materials. ISSN 0021-9983
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Permanent indentation characterization
for low-velocity impact modelling using
three-point bending test
Natthawat Hongkarnjanakul, Samuel Rivallant,
Christophe Bouvet and Arturo Miranda
Abstract
This paper deals with the origin of permanent indentation in composite laminates subjected to low-velocity impact. The three-point bending test is used to exhibit a non-closure of matrix crack which is assumed as a cause of permanent indentation. According to the observation at microscopic level, this non-closure of crack is produced by the blocking of debris inside matrix cracking and the formation of cusps where mixed-mode delamination occurs. A simple physically-based law of permanent indentation, ‘‘pseudo-plasticity’’, is proposed. This law is qualitatively tested by three-point bending finite element model and is lastly applied in low-velocity impact finite element model in order to predict the permanent indentation. A comparison between low-velocity impact experiments and simulations is presented.
Keywords
Permanent indentation, dent depth, impact behaviour, damage mechanics, damage tolerance, three-point bending test
Introduction
In recent years, the use of composite structures has increased in aeronautical and aerospace applications, thanks to their high specific strength and stiffness with weight reduction. However, the brittleness of the composites is reflected in their poor ability to resist impact damage that might be encountered in accidental situations such as tool drops during manufacturing or maintenance, etc. This can cause a severe reduction of strength of the structure both in tension and compres-sion. However, the difficulty to detect internal damage is a more sensitive aspect due to unobvious visible sign on the surface. This has led to a fundamental design concept for damage tolerance of aeronautic structures. To achieve damage tolerance requirement, when the damage does occur but is invisible, composite parts must have sufficient residual strength to resist failure at design ultimate load. As a result, ‘‘barely visible impact damage’’ (BVID) is defined as level at which
the damage is detectable.1In practice, the BVID is
com-monly inspected by measuring ‘‘dent depth’’ or ‘‘per-manent indentation,’’ which is specified as a difference
between the lowest point in the dent and the surface.2
Understanding of permanent indentation mechanism is therefore essential to allow for designing the composite
structures with respect to damage tolerance. In the last decades, a number of researchers have studied damage
due to low-velocity impact3–7; however, the
phenom-enon of permanent indentation is still not very well
understood.4,5,8–12Therefore, the argument of
indenta-tion mechanism is still open.
In literature, some analytical predictions and
inden-tation models have been proposed. Caprino et al.9,10
experimentally studied low-velocity impact on compos-ite laminates and found that if the ratio of impact energy to the penetration energy, which is unaffected by the loading speed, is known, the permanent inden-tation can be evaluated, regardless of laminate type and thickness. The prediction of permanent indentation is accomplished by an exponential equation based on the best-fit method from available experimental data. The question arises whether this approach is valid when laminate configurations/materials are changed. Hence,
Universite´ de Toulouse: ISAE, INSA, UPS, Emac; ICA (Institut Cle´ment Ader), Toulouse, France
Corresponding author:
Natthawat Hongkarnjanakul, ISAE (Institut Supe´rieur de l’Ae´ronautique et de l’Espace); 10 Avenue Edouard Belin, BP 54032, 31055 Toulouse Cedex 4, France.
Email: natthawat.h@gmail.com
0(0) 1–14
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other methods should also be considered such as finite element (FE) analysis. For example, a simulation of low-velocity impact damage and permanent indenta-tion based on continuum damage mechanics, as reported in Ref [5–7]. The permanent indentation mechanism is included in the non-linear shear behav-iour which is decomposed into (1) elastic undamaged, (2) elastic damaged and (3) inelastic damaged compo-nents. Irreversibility of the last term is attributed to unclosed cracks and causes the formation of permanent indentation. These models consist of fracture mech-anics approach with damage evolution. Thus, shear fracture toughness under intralaminar matrix failure needs to be measured. Nevertheless, this value is
diffi-cult to evaluate,13especially out-of-plane shear
behav-iour which is a fundamental direction for the formation
of permanent indentation. As a result, Donadon et al.5
simply assumed out-of-plane shear behaviour to be identical to in-plane shear behaviour for their fabric laminates because of the lack of experimental results for out-of-plane shear behaviour, thus some limitations for this assumption might be of concern.
The value of permanent indentation significantly
increases when fibre failure is detected.11,12 On the
other hand, at a small value of permanent indentation, a small amount of fibre failure is observed. The per-manent indentation is probably driven by plasticity of resin ductility/non-linear shear behaviour and matrix cracking. To model all these complex mechanisms of permanent indentation, i.e. fibre failure, matrix
crack-ing/plasticity, He et al.12 have proposed a model of
permanent indentation associated with anisotropic elasto-plasticity theory in continuum mechanics, but five internal coefficients for the shear model need the trial and error optimization. Excellent predictions of permanent indentation value in function of impact energy for two tested material systems are presented. Also, the importance of fibre failure affecting
perman-ent indperman-entation was also shown in Bouvet et al.’s4
simu-lation on the evolution of impact energy, even if their permanent indentation law is associated with matrix cracking rather than fibre failure. In fact, this law is based on an experimental investigation from Abi
Abdallah et al.8who claimed that the permanent
inden-tation is formed because the debris get stuck inside matrix cracking void (Figure 1). These debris play an important role to block the closure of matrix cracks returning to initial position. The accumulation of these opening of matrix cracks in each ply can contrib-ute to the formation of permanent indentation. Moreover, during an impact event, fibre failure can locally degrade the plies’ rigidity and cause a great opening of these matrix cracks and delamination. This can indirectly affect the permanent indentation to be more significant. Thanks to this assumption,
Bouvet et al.3,4are able to present accurate simulation
results for their test cases, for example the indentation value, and deformed shape of the plate. However, two necessary parameters in their simulation i.e. debris size and debris stiffness are assumed and not yet identified. Due to lack of explanation on physical basis of per-manent indentation in literature, the aim of this current paper is to clarify this point in order to improve the permanent indentation response on low-velocity impact
simulation3,4and to propose a way to identify
param-eters governing permanent indentation. Thus, a three-point bending test is chosen for this study, which enables the creation of matrix cracks and follows their evolution: opening, closure and eventual blocking of closure. Mechanism of crack opening/closure par-ticularly in out-of-plane direction is highlighted, which can lead to new interpretation and understand-ing of the formation of permanent indentation. A simple law of permanent indentation is presented, and a single required parameter is proposed, based upon experiment-model identification. The final step is to verify this indentation law on impact model and validate on test cases.
Experimental
Material and test setup
Three-point bending tests are conducted under hydrau-lic machine at a displacement rate of 0.3 mm/min.
Specimens with a dimension of 50 10 mm2
(length width) are made of T700/M21 (carbon-epoxy). Two stacking sequence configurations of speci-mens are detailed in Table 1, which particularly enable the observation of matrix cracking in the internal plies
0° 90° 0°
impactor debris
Open blocked delaminations
Figure 1. The principle of creation of permanent indentation.3
Table 1. Specimen configurations.
Name Layups Thickness, h (mm)
I1 [0, 906, 0] 2.0
I2-A [02, 902, 02] 1.5
I2-B [02, 902, 02] 1.5
oriented at 90. To predominantly generate the matrix
cracking at the 90 plies (generally coupling with
delamination) due to shear with less possibility to
have tensile or compressive fibre failure of 0 external
plies, the span (distance between the two lower
sup-ports) to thickness ratio (L/h) should be reduced.14
Thus, 10 mm of span and three cylinder supports of 6 mm of diameter are chosen for this study (Figure 2). During the three-point bending test, a digital camera with macro lens meanwhile captures in the section of specimen being affected with an automatic interval timer function. These images are then post-analysed using image correlation technique in order to observe the translation of deformed specimens and their cracks. Since the physical basis of the mechanism of permanent indentation is initially assumed due to the debris which
may block the crack closure,8 crack void and debris
should be observed in optical microscope. Therefore, each specimen must be polished before test to ensure that the debris observed does not come from small par-ticles during polish process (generally done after test). Moreover, areas of interest are photographed by scan-ning electron microscope (SEM) in order to observe more precisely the debris.
Three-point bending test and local deformation
As a result of reducing the span in three-point bending
test to favour the crack in 90 plies, consequent
influ-ences must be taken into account i.e. relatively high shear effect, and high local contact pressure of the spe-cimen against cylinder supports, known as ‘‘local deformation’’ (Figure 3(a)). In particular, if the speci-men has small span and large thickness, the local deformation effect will be very significant comparing
to shear and bending effects15 (Figure 3(b)). In order
to later analyze the three-point bending from FE model (cf. section §3) which does not take into account the local deformation, and highlights shear and bending behaviour, the effect of local deformation should be minimized as much as possible.
The displacement obtained from the machine can be
defined as the global displacement, global. This quantity
can be subdivided into two components namely the
bending displacement, bending, where shear effect is
also included, and the local deformation, local.
global¼ bendingþ local: ð1Þ
(a) (b) 2 mm Local deformation Local deformation B1 L h B2 A1 A2
Figure 3. Demonstration of local deformation effect on three-point bending test (a) present test on specimen I1 and (b) relative influence of local deformation and shear effects with respect to bending.15
Spotlight
Digital camera with macro lens
Hydraulic machine three-point bending supports Specimen and
L = 10 mm specimen
∅ 6 mm
h
The bending displacement cannot directly be mea-sured but it can be evaluated if the global displacement and the local deformation are known (equation 1). Thanks to the image correlation technique in 2D, the global displacement can be determined by monitoring the movement on the cylinder supports. Also, the local deformation can be simply evaluated by measuring the different displacement between cylinder contact noses and neutral axis of the specimen, at points A1, A2, B1 and B2, as shown in Figure 3(a).
local¼ j A1ÿ A2j þ j B1ÿ B2j ð2Þ
where A1, A2, B1 and B2 are the vertical
displace-ments at point A1, A2, B1 and B2, respectively. The software then interprets the difference from the series of images and gives the results of displacements in pixel units; these displacements can consequently be converted to mm. Nevertheless, the acquisition frequen-cies for global displacement – obtained from hydraulic machine – and for image correlation are different. Thus, the displacement measures from image correlation are fitted by a cubic polynomial interpolation (Figure 4(a)). Lastly, the bending displacement is obtained, as shown in Figure 4(b). Hereafter in this paper, unless otherwise specified, the term ‘‘displacement’’ means the bending displacement rather than the global displacement.
Opening and closure of matrix crack
The evolution of crack opening/closure is studied according to previous observation of formation of
per-manent indentation proposed by Abi Abdallah et al.8
Similarly to what occurs in an impact event including a loading and a rebound, three-point bending test is car-ried out with only one cycle of loading and unloading. Many tests were performed, but unfortunately, it is quite complex to control the formation of a unique crack in the specimen, to avoid uncertain crack loca-tion, or even to avoid fibre rupture under the indentor.
Only specimens with a single crack and no fibre rupture were analysed, so that only the influence of matrix cracks could be studied. Finally, only four interesting specimens are chosen. The results from three-point bending test of specimen I1 are presented in Figure 5.
The specimen is loaded until the appearance of a 45
matrix cracking (normally coupling with delamination), which can be noticed due to the dramatic load drop at state 2–3. When the matrix crack occurs, the specimen’s internal rigidity suddenly decreased but the global dis-placement remains the same. As a consequence, the local deformation is released and leads the bending dis-placement shift at state 2–3 (see also the disdis-placement
relation in Figure 4(b)). The out-of-plane crack width z
is also measured by using image correlation technique (same technique as determining the local deformation). Unloading is then applied and the crack width gets smaller (state 3-4-5). At the end of the test (point 5), one can notice that there is a residual crack opening when the force is null. This phenomenon is assumed to be the basis of the formation of permanent indentation.
If this unclosed crack leads to the formation of per-manent indentation, it is questionable as to what mech-anisms influence the Residual crack opening width. Three additional tests on configuration I2 are conse-quently performed but different load cases are intro-duced. The load is continuously applied after the onset of crack initiation and before the presence of fibre failure to three different levels of added displacements, i.e. 0.02 mm, 0.08 mm and 0.15 mm for specimens I2-A, I2-B and I2-C, respectively (Figure 6(a)). The results exhibit some scatters in the onset of crack formation data for these tests, but different crack widths are inves-tigated according to the different added displacement (Figure 6(b)). There is no fixed value of the residual crack opening, but it probably depends on the maximum crack opening.
Evolution of crack opening/closure both in
trans-verse direction t, and in out-of-plane direction z,
0.0 0.1 0.2 0.3 0.4 0 20 40 60 80 100 120 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.10 0.20 0.30 L o ca l d ef o rm at io n ( m m ) Global displacement (mm) Camera / load ( ) Time (sec) D is p la ce m en t (m m ) Global displacement (δglobal )
Local deformation (δlocal
) Bending displacement (δbending
) (a) (b) Load ( ) Unload ( ) 4 3 2 2 3 1( ) c( ) c( ) c
c global blobal global
local= + + + δ δ δ δ 4 3 2 2 3 1( ) c( ) c( ) c
c global blobal global
local= δ + δ + δ +
δ
Camera / unload ( ) Cubic poly. / load ( ) Cubic poly. / unload ( )
Figure 4. Displacement output data of specimen I1 (a) local deformation in function of global displacement from image correlation technique and cubic polynomial estimation and (b) different types of displacement during three-point bending test.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.1 0.2 0.3 0.4 I2-A I2-B I2-C 0.0 0.4 0.8 1.2 1.6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 I2-A I2-B I2-C 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 I1 ( ) I1 ( ) I2-A ( ) I2-A ( ) I2-B ( ) I2-B ( ) I2-C ( ) I2-C ( ) F o rc e (k N ) Displacement (mm) Displacement (mm) C ra ck o p en in g /c lo su re in z -d ir ec tio n ( m m ) (a) + 0.08 mm (b) + 0.15 mm + 0.02 mm Load ( ) Unload ( ) Displacement (non-dimension) Crack opening/closure (non-dimension) (c) 20% - 40% Load ( ) Unload ( ) z t l δ t δ t δ t δ t δ z δ z δ z δ z
Figure 6. (a) Force-displacement for tests of specimens I2-A, B and C; (b) crack opening/closure for tests of specimens I2-A, B and C and (c) non-dimensional crack opening/closure for tests of specimens I1, I2-A, B and C.
C ra ck o p en in g /c lo su re , δz ( m m ) F o rc e (k N ) 0.00 0.40 0.80 1.20 1.60 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.05 0.10 0.15 0.20 0.25 Displacement (mm) Displacement (mm) 1 2 3 4 5 1 2 3 4 5 5 4 3 2 1 load unload load unload 2 mm z t l δz δt
is determined at the centre of the matrix crack in the thickness direction of the specimen; the subscripts t and
z denote transverse and out-of-plane directions,
respectively, as demonstrated in Figure 5. The results
of non-dimensional crack evolution, i.e. t=maxt and
z=maxz , reveal a sensible convergence for all tests.
That is, using non-dimensional values of imposed dis-placement and crack closure, the shape of crack clo-sure-displacement curves for all tests are similar, and regardless of how big the maximum crack opening is, the final crack closure returns to around 20%–40% of maximum crack opening (Figure 6(c)) for both crack directions (t- and z-directions) and for both specimen configurations.
In order to investigate the cause of this unclosed crack, a microscopic observation is performed. The
micrographs from Figures 7(a) and 8(a) show the 45
crack appearance, and the coupling between matrix cracking and delamination for specimens I1 and I2-B, respectively. These cracks are observed at higher mag-nification by SEM in order to more precisely observe the crack. The debris inside the crack are clearly visible
in Figure 7(b) for specimen I1 and to a lesser extent for specimen I2-B (Figure 8(b)). One must keep in mind that debris may be hidden inside the matrix cracking and are not always clearly visible from microscopic observation.
Detailed micrographs of specimen I2-B in Figure 8(c) and (d) show another mechanism leading to the non-closure of cracks. Figure 8(d) (lower interface) shows
the classical formation of cusps16 in the ply interface,
due to a mixed mode delamination. In this figure, the interface of the cusps is not entirely opened, and the upper and lower plies are still linked together by packs of resin. Looking further from the delamination tip (Figure 8(c), upper interface), the cusps are completely formed and can be assimilated to debris. Both types of cusps (entirely or partially formed) will prevent the clos-ure of delamination cracks to initial position.
According to experimental observations, we found that the main mechanisms leading to permanent inden-tation are both the accumulation of debris inside matrix cracks and the presence of cusps inside delamination interfaces. Moreover, measurements of the opening/
2 mm Matrix cracking Delamination 902 02 02 (d) 100 µm 100 µm (a) (b) 100 µm (c)
Figure 8. Post mortem observation (a) micrograph of specimen I2-B after test; (b) scanning electron microscope (SEM) observation showing a void due to matrix cracking; (c) and (d) cusps induced by delamination.
2 mm Matrix cracking 906 0 0 Delamination Debris (a) (c) (b) 20 m 100 µm
Figure 7. Post mortem observation (a) micrograph of specimen I1 after test and (b) scanning electron microscope (SEM) obser-vation of debris inside matrix cracks; (c) SEM obserobser-vation of delamination.
closure of cracks show that the final crack width depends on the maximum width reached during open-ing of the crack in the test.
Modelling of permanent indentation
Permanent indentation hypothesis
Experiments in the proceeding section demonstrate that, under unloading, different microscopic mechan-isms in matrix cracks (debris) and delamination inter-faces (debris, cusps) result in a global phenomenon of non-closure of the matrix cracks. From a meso-scale point of view, the final matrix crack width is found proportional to the maximum crack width reached during the test. It leads to the definition of the crack closure coefficient , expressed as follows:
0¼ max ð3:1Þ
with
maxð Þ ¼ max
t ð ð ÞÞ ð3:2Þ
where 0is the final crack width, after unload, and max
is the maximum crack width ever reached during the crack opening. t and are respectively the total time and the current time. The relation in equation (3.1) is a fundamental relation to drive the formation of perman-ent indperman-entation, called ‘‘pseudo-plasticity’’ model. This law physically comprises both the blocking of debris and resin plasticity in plies. It can independently be rewritten for each crack direction: transverse and out-of-plane, without taking into account the compressive transversal direction. 0t ¼ t maxt if t 0 ð4:1Þ 0z¼ z max z if z 0 z minz if z50 ð4:2Þ where the 0
t, 0z are the final crack-closure width in
transversal and out-of-plane directions, respectively. Furthermore, the crack closure coefficient from both directions in this study is assumed to be identical
(t¼ z) due to the fact that the matrix cracks are at
an angle close to 45, which is confirmed by the results
shown in Figure 6(c).
The crack evolutions are physically observed in micro-scale in terms of debris and formation of cusps, but the proposed law (equations 4.1–4.2) is responded at meso-scale in the proposed model. These law and the findings of the out-of-plane non-closure crack from previous section are comparable to the non-linear
out-of-plane shear damage model proposed by
Donadon et al.,5 as shown in Figure 9. That is, the
residual opening component is about 30% of maximum deformation component.
Validation of permanent indentation model
on three-point bending test
The objective of this section is to qualitatively test the proposed indentation law with a three-point bending FE model, and understand the interaction of crack opening/closure on the particular discrete model. Mesh types and material laws/properties are based on the impact model from Ref [3,4] in which three damage types are considered namely fibre failure, matrix crack-ing and delamination, as shown in Figure 10. Dynamic explicit analysis is also maintained with a reasonably slow velocity (0.1 mm/s) to eliminate dynamic problem. Since the three-point bending model is created based on the impact model, after the validation on the three-point bending model, the permanent indentation law can directly be applied to the impact model.
The experimental investigation in the previous sec-tion showed that crack non-closure is due to both matrix cracking and delamination; but at meso-scale, it can be considered as a non-closure law only applied in the matrix cracks. The permanent indentation law is integrated inside vertical interface elements where matrix cracking is assigned. To identify the damage that occurred from experiment test, only one row of interface elements is allowed to possess matrix cracking
with the purpose to simulate the 45 matrix cracking
(cf. Figure 13(a)–(b)). Thus, the element size was
chosen according to location of the presence of 45
matrix cracking and to respect cohesive zone model for delamination. The element size for three-point bending model is about three times smaller than the reference low-velocity impact model in Ref [3,4], whereas the element’s thickness is maintained the same. Figure 11 shows the mechanism of crack
Load ( )
Unload ( )
Irreversibility ≈ 30% of max.strain
Figure 9. Non-linear in-plane and out-of-plane shear damage model from Donadon et al.5
opening/closure that corresponds with the permanent indentation law in vertical interface elements.
Initiation of matrix cracking. Before reaching the onset of matrix cracking, the behaviour of vertical interface is purely elastic: the interface elements connect their neighbouring volume elements by the interface stiffness K (assumed to be very high: 500,000 MPa/mm, given in Table 2), defined in three directions:
t¼ K t ð5:1Þ
lt¼ K l ð5:2Þ
zt¼ K z ð5:3Þ
The initiation of crack is governed by matrix tensile failure (equation (6)) based upon Hashin’s criterion. This criterion is assigned into neighbouring volume elements, but responded in interface elements.
Vt þ tf !2 þ V lt ÿ 2 þ V tz ÿ 2 f ÿ 2 1 ð6Þ where V
t , ltV, tzVdenote the stresses calculated in
neigh-bour volume elements. tf the failure transverse tensile
stress and f is the failure in-plane shear stress. The
essential mechanical characteristics of T700/M21 are given in Table 2. The purpose of this three-point bend-ing model is to validate the crack openbend-ing/closure simu-lation. The failure stresses are modified to overcome the
K t
δ
zδ
maxδ
K( )
δ size crack 0 δ (i) Before crack S tr ess Load (crack opening)
Unload (crack closure) t
l z Failure evaluated in volume elements
(ii) After crack
Figure 11. Permanent indentation law representing matrix cracking interface elements. Figure 10. Three-point bending model and element types.
possible dispersion of rupture value in experimental tests, so that the cracks can appear at the same time for both experiment and simulation. For example, the
values of tf and f, for material T700/M21,17
which equal to 60 MPa and 110 MPa, respectively (Table 2), are replaced by 39 MPa and 78 MPa for modelling case of I2-B. Moreover, another reason of the modified fail-ure stresses is that the three-point bending model has very thick finite elements comparing to the total thick-ness. As a result, the estimation of shear distribution in through-thickness direction is inaccurate and
compli-cated to predict18; but this problem is not evident for
the reference impact model in Ref [3,4] because of a relatively thin volume element comparing to the lamin-ate thickness. However, a modification of reducing element’s thickness or adding number of element was not made in order to keep the similarity to impact model and the objective of parameter’s identification. Opening and closure of cracks. Once the matrix cracking criterion in volume elements (equation (6)) is reached (the initiation of crack), the adjacent interface elements lose their rigidity, their stresses in three directions turn to zero and the crack instantaneously opens (not taking into account the energy for propagation). Also, the delamination is automatically linked, which is per-formed by a conventional criterion of fracture mech-anics on the horizontal interface elements. Then, after initiation of crack, the crack opening is only gov-erned by delamination, as far as the crack width is increasing.
When the crack tends to close, during unloading, the crack closure law – presented in Figure 11 – is applied only at the interface elements for matrix cracking. As long as the crack width is larger than the final crack
closure widths, 0
t and 0z, expressed previously, the
crack freely closes with no resistant stress. But if the crack closure goes below these thresholds, the rigidity will be applied to prevent the crack closure in the z- and
t- directions. The implementation of the crack closure
law can be given as:
t ¼ 0 if t40t K tÿ 0t ÿ if t 0t ( ð7:1Þ lt¼ 0 ð7:2Þ zt¼ 0 if j j 4 z 0z K zÿ 0z ÿ if j j z 0z ( ð7:3Þ with 0
t and 0zobtained in equations (4.1) and (4.2). The
finding from all experiment tests in Figure 6(c) shows the final crack width is about 20%–40% of the max-imum crack opening. Therefore, an average value of 30% is taken as average crack closure coefficient, that
is, t ¼ z¼ 0:3.
The validation of model of three-point bending is done for all four test cases and gives similar agreement; but only the case I2-B is presented here because its con-figuration is similar to the one of impact layup. Since the failure stresses are set to break as same as in experi-ment, good onset of load drop is automatically found, as presented in force-displacement curves in Figure 12(a), even if its magnitude is overestimated. This overestimation is due to delamination, that is, the end of lower interface delamination in experiment
stops at about the lower support’s contact
(Figure 13(b)), whilst the simulated delamination goes further, as represented by the damage variable in Figure 13(a). The ratio of the energy release rate to the critical energy release rate between two modes
(GI=GIc and GII=GIIc) is considered, we found that
this overestimation is predominated due to shear mode (mode II), which is supposed due to:
– In practice, at the contact between the supports and specimen there is relatively high compressive
pressure affecting inside laminate, therefore GIIc
should be increased to reduce the delamination
effect in mode II19; but this influence is not
taken into account in the present model.
– The friction effect at the interface for
delamin-ation7has not been included in model.
The displacement is continued after the onset of crack
formation and the maximum crack opening (maxt and
maxz ) is correctly predicted (Figure 12(b)). During
unloading, the numerical results also exhibit a good accordance in returning of crack width to 30% Table 2. Mechanical characteristics of T700/M21 unidirectional
ply for the three-point bending model.17
Ply density 1600 kg/m3
Ply thickness 0.25 mm
Etenl Tensile Young’s modulus in fibre direction
130 GPa Ecompl Compressive Young’s modulus
in fibre direction
100 GPa
Et Transverse Young’s modulus 7.7 GPa
Glt Shear modulus 4.8 GPa
lt Poisson’s ratio 0.33
ft Tensile failure stress in transverse direction
60
f In-plane shear failure stress 110
K Interface element stiffness
of matrix cracking
500000 MPa/mm
of maximum, that is, the final crack-closure width (0
t
and 0
z) when the load is totally released.
The three-point bending model presented is a rather qualitative study in order to understand the mechanism of crack opening/closure. The key idea of the proposed law is related to the residual opening of crack as explained above. To simulate the permanent indenta-tion in actual practice, the validaindenta-tion on low-velocity impact model of laminated plate is necessary. Similar to the three-point bending model, the residual opening of each element and each layer is gathered, so the perman-ent indperman-entation of laminated plate can be formed. However, some differences of their characteristics must be concerned, for example, matrix cracks for three-point bending are free edge, whilst for impact it is internal (closed boundary condition). Therefore, veri-fication of crack closure coefficient on low-velocity impact model is needed.
Verification of permanent indentation
law on low-velocity impact modelling
To complete the goal of the study, it is essential to validate the indentation law and crack closure
coeffi-cient on low-velocity impact model. The new
indentation law is introduced in the existing core of impact model in Ref [3,4]. The permanent indentation concept incorporates with matrix cracking interface in the discrete impact model is still remained. That is, the accumulation of opening of matrix cracks in each ply can contribute to the formation of permanent indenta-tion. Fibre failure also indirectly affects the permanent indentation to be more significant since the fibre failure plies near impact point locally lose their rigidity and induce a great opened block of matrix crack.
Experimental study of low-velocity impact was car-ried out. T700/M21 laminated (same material as
three-point bending specimens) of 150 100 4 mm3plates
were fabricated and tested on eight case studies with six different stacking sequences and three impact energies, as detailed in Figure 14. The layups are eight-double-ply, mirror-symmetric, quasi-isotropic, which are made, for this study, in order to facilitate impact damage observation and reduce calculation time. Then, the optimized layup will be the subject of further study after this work is accomplished.
In experiment, permanent indentation of impacted plates is determined by measuring the difference between the lowest point in the dent and the plate
sur-face.2Due to twisted form and non-uniform plate
sur-face (Figure 15(b) to (c)), it is necessary to give a definition of surface plate, especially when a precision of permanent indentation is needed. In this study, a fixed distance at about 1.5 of indentor radius, which
0.00 0.40 0.80 1.20 1.60 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00 0.10 0.20 0.30 0.40 0.00 0.10 0.20 0.30 0.40 Exp -u Exp -v Model -u Model -v max t δ max z δ 0 t δ 0 z δ Force (kN) Displacement (mm) Displacement (mm) Crack opening-closure (mm) Model Experiment (a) (b) δ t δ t δ z δ z
Figure 12. Experimental validation of (a) force-displacement response and (b) crack opening/closure for case study I2-B.
0.0 0.2 0.4 0.6 0.8 1.0 11 12 13 14 15 16 17 18 19 20 IIc IIG GI/ Ic G G/
Location along specimen’s length
1. Upper interface 2. Lower interface
Delamination 1 mm Exp. (b) (a) 1 mm Model Delamination 1. 2. Damage variable Mode I mode II Damage variable
Figure 13. Delamination model of specimen I2-B (a) illustra-tion at maximum crack opening with damage variable and vari-ation of failure mode with respect to locvari-ation along specimen’s length and (b) delamination’s length from experiment observed by scanning electron microscope (SEM) compared to model.
is approximately around the dent, is selected to be a representation of average plate surface. Then, perman-ent indperman-entation can be measured by image correlation technique in 3D.
In order to validate the permanent indentation law, a parametric study of the crack closure coefficient (), the single required parameter, was carried out. Figure 14 gives the indentation results of eight test cases which is a comparison between the experiments and the simulations. The influence of tuning the crack closure coefficient is shown, that is, this parameter dir-ectly affects the permanent indentation. The parameters
t¼ z¼ 0:35 instead of 0.30 from previous section
were found to obtain good agreement between experi-ments and simulations on variation of stacking sequence and impact energy. The different error of the worst case is 24%, which seems acceptable since experimental indentation may be uncertain due to e.g.
the scattering of experimental data,12relaxation of
per-manent indentation after 48 hours8 or the method of
measurement as mentioned above. However, the ten-dency of permanent indentation is well captured.
The result of example case E2 [-452, 452, 02, 902]sat
25 J is showed in Figure 15, demonstrating that the proposed law is able to predict:
. Value and shape of permanent indentation on
impacted side (Figure 15(a)–(c)).
. Ply-splitting on non-impacted side due to matrix
cracking is simulated, as shown in Figure 15(d)–(e)
and No.10–No.100in Figure 16, in spite of an
over-estimation of the simulation (red zone in Figure 15(e)).
. This law also allows the shape of the deformed plate
after impact to be obtained as shown by the twisted shape (Figure 15(b)–(e)). The law also allows the creation of residual opening of delamination, according to the interaction with other failure modes, as can be seen from the micrograph in Figure 16(a)–(b). The same sub-numbers indicate the similarity between experiment and simulation at the particular locations.
Although the overestimation of opened delamination
in the middle of the layup has been found
(Figure 16(b)), it does have small effect on the manent indentation on impacted side since the per-manent indentation is due to the gathering of residual openings of three or four plies on the impacted side as shown in Figure 16(b).
One can notice that the middle of the layup E2 is [02,
904, 02], which is similar to the layup of specimens from
three-point bending test I1 [0, 906, 0] or I2 [02, 902, 02].
The 45 matrix cracking at locations 2, 3 coupling with
delamination at locations 6, 7 in Figure 16(a) from
0 0.1 0.2 0.3 0.4 0.5 C1 C2 D1 D2 E1 E2 E2 E2 Experiment Mo d el = 0 .3 5 M o d el = 0 .3 0 30J 17J 25J La y u p s/ im p act ener gy [02 ,9 02 ,4 52 ,-4 52 ]s [9 02 ,02 ,-4 52 ,4 52 ]s [4 52 ,02 ,-4 52 ,9 02 ]s [-4 52 ,9 02 ,4 52,0 2] s [4 5 2,-4 52,9 02 ,02 ]s [-4 52 ,4 52 ,0 2,9 02 ]s [-4 52 ,4 52 ,0 2,9 02] s [-4 52,4 52,0 2 ,9 02 ]s Permanent indentation (mm) µ µ F ig u re 1 4 . Comparison of permanent indentation on eight test cases, var ying la yups and impact energies.
impact test seems comparable to micrographs from three-point bending tests in Figure 7(a) or Figure 8(a). This can confirm that the choice of using three-point bending test to characterize permanent indentation is reasonable.
Assigning the non-closure crack law into only matrix cracking interfaces is able to contribute not only to the permanent indentation but also the residual opening of delamination. On the contrary, as the present model allows the coupling between matrix cracking and delamination, giving the non-closure crack law at delamination interfaces also seems realistic.
The local plasticity under the indentor is also important for the permanent indentation especially for thick laminates. This topic is currently in progress and will shortly be combined to the global impact model. Thus, the crack closure coefficient may be later adjusted. This point must be studied further.
Conclusion
In the presented paper, a model of permanent indenta-tion due to low-velocity impact was proposed using three-point bending test. An experimental observation
of matrix crack opening/closure leads to the idea of permanent indentation formation. When a matrix crack appears, it usually does not completely close and does remain at approximately 30% of maximum crack opening, observed in both transversal and out-of-plane directions. This unclosed crack is due to the blocking of debris inside matrix cracking and cusp for-mation due to mixed mode (peel and shear) delamin-ation. By taking into account both findings, a simple material law of indentation was proposed according to the unclosed crack which is defined by the crack closure coefficient.
The pseudo-plasticity law was initially tested with three-point bending model based upon a selected case study (I2-B) with the aim of understanding how the crack opening/closure mechanism interacts to the dis-crete model. Consequently, the same law was validated on low-velocity impact model, and the crack closure coefficient was found to directly affect the value of per-manent indentation.
Advantages of the proposed law are its simplicity, and need of only a single parameter which can be found from numerically adjusting the parameter. Also, this law can predict deformed shape of impacted plate
0.0 0.1 0.2 0.3 0.4 0 15 30 45 60 75 90 105 120 135 150 Exp Model Impacted side Experiment Permanent indentation (mm)
Location along specimen’s length(mm)
Permanent indentation
Permanent indentation
cut A-A cut A-A
cut A-A (exp.)
cut A-A (model) Reference for indentation measurement
(∼1.5Rimpactor from impact point)
0 mm 0.4 mm Permanent indentation Impacted side Non-impacted side 0° 90° 100 mm Model (a) (d) (e) Non-impacted side 0 mm 1.0 mm (b) (c) Ply-splitting 100 mm
F ig u re 1 6 . Comparison of longitudinal section cut cr ossing impact point after impact for impact test case E2 [-45 2 , 45 2 , 02 , 90 2 ]s at 25 J (a) micr ograph fr om experiment; (b) simulation.
such as twisted form, or opened block of delamination. This benefit is essential for CAI simulation in next step when buckling failure is partially due to plate deflec-tion. On the other hand, it still has some limitations, for example, at high-impact energy with large permanent indentation or penetration, the indentation is mainly governed by the fibre failure, and thus supplementary criterion must be added. However, if the objective is to numerically optimize composite structure at BVID,
approximately 0.3 mm–0.6 mm of dent depth,1the
pro-posed hypothesis is justified. Acknowledgements
This work was granted access to the HPC resources of CALMIP under the allocation 2012-P1026.
Funding
The authors gratefully acknowledge the partial financial sup-port through the THEOS Operation Training Program (TOTP).
Conflict of Interest None declared. References
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